Form of Growing Strings

Patterns and forms adopted by nature are often the results of simple dynamical paradigms. Here we show that a growing self-interacting string attached to a tracking origin, modeled to resemble nascent polypeptides in vivo, develops helical structures which are more pronounced at the growing end. We also show that the dynamic growth ensemble shares several features of an equilibrium ensemble in which the growing end of the polymer is under an effective stretching force. A statistical analysis of native states of proteins shows that the signature of this nonequilibrium phenomenon has been fixed by evolution at the C terminus, the growing end of a nascent protein. These findings suggest how evolution may have built on the properties of a generic nonequilibrium growth process in favoring helical structures in nascent chains.

Figure 1

Conformations of a growing chain for various $r_{\mathrm{growth}}$ as indicated ($\tau$ is defined in the text). The polypeptide chains shown are all of 30 residues. The chain is grown from the left end and the growing end is kept fixed. The growth rates decrease from top to bottom. In a rather wide range of intermediate growth rates [(b) and (c)] an ordered helix is formed at the growing end and another one, more distorted and equilibrated, appears at the other end. ${\tau }_{\mathrm{relax}}$, at the growth temperature, is estimated to be $\sim 80\tau$.

Figure 2

Left, schematic representation of two single molecule stretching experiments that we discussed: (a) fixed force mode, (b) fixed stretch mode. Right, sketch of the phase diagram at $T=0$. The minima were found with molecular dynamics and Monte Carlo simulations at low $T$ and varying force [in (a)] and end-to end distance [in (b)]. In (b), the velocity is small so that the string is always in quasiequilibrium. Most of the stable states in Fig. 2b are characterized by a coexistence of two states found in Fig. 2a. $N$ refers to the length of the polymer.

Figure 3

(a) Propensity of forming helices and strands as a function of residue position away from the C terminus (dashed line, blue online) and N terminus (solid line, red online). The data are computed based on a PDB set of 291 nondisordered protein structures with CO smaller than 0.15 and with the secondary structures assignment given in the PDB files. To determine a propensity at a given position we only took proteins with length greater than double the position number, in order to avoid interference from the opposite end. (b) The amino acid concentrations in 20-residue fragments associated with the N (solid line, red online) and C terminus (dashed line, blue online) computed from the same set of proteins used in the top figure (amino acids are labeled with the one letter code).